Engineering Mathematics

Vector Calculus

Gradient, divergence, curl and directional derivatives — how scalar and vector fields change in space.

PART 1

Topic Breakdown & Traps

Vector calculus describes how fields vary through space. The gradient

\nabla f

of a scalar field points in the direction of steepest increase and its magnitude is that rate of increase. The divergence

\nabla \cdot F

of a vector field measures the net 'outflow' from a point (a source if positive, a sink if negative). The curl

\nabla \times F

measures local rotation. The directional derivative projects the gradient onto a chosen *unit* direction to give the rate of change along it.

Gradient of a scalar

f

\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})

.

Divergence of

F = (F_{1}, F_{2}, F_{3})

\nabla \cdot F = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z}

(a scalar).

Curl:

\nabla \times F = (\frac{\partial F _{3}}{\partial y} - \frac{\partial F _{2}}{\partial z}, \frac{\partial F _{1}}{\partial z} - \frac{\partial F _{3}}{\partial x}, \frac{\partial F _{2}}{\partial x} - \frac{\partial F _{1}}{\partial y})

(a vector).

Directional derivative of

f

along unit vector

\overset{u}{^}

D_{\overset{u}{^}} f = \nabla f \cdot \overset{u}{^}

⚠
Divergence is scalar, curl is vector. Returning a vector for divergence (or a scalar for curl) is a category error that loses marks instantly.
⚠
Normalise the direction. The directional derivative uses a *unit* vector. Forgetting to divide by $∣ u ∣$ inflates the answer — e.g. direction $(1, 1, 1)$ must be divided by $3$ .
⚠
Gradient acts on scalars, divergence/curl on vectors. Don't take the gradient of a vector field or the divergence of a scalar.

PART 2

Q1BASIC1 Mark · MCQ

The divergence of

F = (x^{2}, y^{2}, z^{2})

evaluated at the point

(1, 1, 1)

is:

Q2MEDIUM2 Marks · NAT

For

f = x^{2} + y^{2} + z^{2}

, the magnitude of the gradient

∣\nabla f ∣

at the point

(1, 2, 2)

is ______. (Round off to two decimal places.)

Q3HARD2 Marks · NAT

The directional derivative of

f = x y z

at the point

(1, 1, 1)

in the direction of the vector

(1, 1, 1)

is ______. (Round off to two decimal places.)